Mathematics > Combinatorics

Abstract: A generalized quadrangle is a point-line incidence geometry $\mathcal{Q}$
such that: (i) any two points lie on at most one line, and (ii) given a line
$\ell$ and a point $P$ not incident with $\ell$, there is a unique point of
$\ell$ collinear with $P$. The finite Moufang generalized quadrangles were
classified by Fong and Seitz (1973), and we study a larger class of generalized
quadrangles: the \emph{antiflag-transitive} quadrangles. An antiflag of a
generalized quadrangle is a non-incident point-line pair $(P, \ell)$, and we
say that the generalized quadrangle $\mathcal{Q}$ is antiflag-transitive if the
group of collineations is transitive on the set of all antiflags. We prove that
if a finite thick generalized quadrangle $\mathcal{Q}$ is antiflag-transitive,
then $\mathcal{Q}$ is either a classical generalized quadrangle or is the
unique generalized quadrangle of order $(3,5)$ or its dual.